0
votes
0answers
11 views

Completeness of a Riemannian manifold with boundary

I have some issues understanding the notion of completeness of a Riemannian manifold with boundary. In the case of Riemannian manifolds without boundary, I found that completeness is usually defined ...
6
votes
1answer
81 views

Which textbook of differential geometry will introduce conformal transformation?

Which textbook of differerntial geometry will have these formulas about conformal transformation? $$\tilde g_{ij} = e^{2\varphi}g_{ij}$$ $$\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ ...
3
votes
1answer
43 views

Klein Bottle Embedding on $\mathbb{R}^4$.

First of all, I am aware of the question in How to embed Klein Bottle into $R^4$ , which was inconclusive. Anyway, I've made some progress, but I still have a question. I am using Do Carmo's ...
1
vote
0answers
13 views

Scalar Curvature of a metric on the hemisphere, from a paper on the Min-Oo Conjecture

I'm reading a paper on the Min-Oo Conjecture (http://arxiv.org/abs/1004.3088), and I'm stuck on the following step in a proposition: Given a metric $g_0(t)$ on the upper hemisphere $\mathbb{S}^n_+$, ...
4
votes
1answer
37 views

Schwarzschild metric tensor normal vectors

The Euclidean Schwarzschild metric describing a manifold (a black hole, though this is not relevant to the question) is given by, $$\mathrm{d}s^2 = \left( 1-\frac{2GM}{r}\right)\mathrm{d}\tau^2 + ...
3
votes
1answer
50 views

The Lie derivative by an infinitesimal action is invertible at an isolated zero point

Let a compact Lie group $G$ act on a manifold $M$. Fix $X \in \mathfrak g$ and we write $X_M$ for the infinitesimal action of $X$. Assume that $p \in M$ is a zero point of $X_M$. Define a linear map ...
3
votes
1answer
25 views

Chain rule quesition: proving that the Weingarten map is self-adjoint

I'm reading through the proof in this paper (http://www.math.leidenuniv.nl/scripties/JaibiBach.pdf) but I'm stuck at the line: "Using the chain rule we get: $L_p(\phi_v) = -Dn(\phi_v) = - \frac ...
3
votes
1answer
64 views

Tensor Laplacian

For a general tensor $T_{\mu_1 \dots \mu_n}$ on a (pseudo-)Riemannian manifold, is it true that $\Delta (T_{\mu_1 \dots \mu_n})= (\Delta T)_{\mu_1 \dots \mu_n}$? In general, it is not true that ...
1
vote
1answer
57 views

A lemma is John Lee's Riemannian Manifold having problem with proving it

The tangential connection on an embedded submanifold $M ⊂ R^n$ is symmetric. the hint is let $X,Y$ be vector fields that are tangent of M at points of M, so is $[X,Y]$ I start with $$T(X,Y)=\nabla_x ...
1
vote
0answers
24 views

Volume form for $SE(n)$ and/or $E(n)$. [duplicate]

I wonder what happens when you construct the Tiling spaces considering the natural action of $SE(n)$ or $E(n)$ rather than $\mathbb R^n$. In order to do that, I need to understand both the riemannian ...
1
vote
0answers
38 views

Is $\nabla_{X}:\Gamma(E) \to \Gamma(E)$ continuos aplication?

Be $\Gamma(E)$ smooth sections space of an vector bundle, $\Gamma(E)$ with Fréchet topology. Gived a conexion in E and a smooth vector field $X$, Is $\nabla_{X}: \Gamma(E) \to \Gamma(E)$ continuos ...
0
votes
1answer
33 views

Orthogonal connection on tangent bundle

What does orthogonality of connection mean in coordinate way? As I understand, a connection $\nabla: \Lambda^1M \rightarrow \Lambda^1M \otimes \Lambda^1M$ is torsion-free iff in any local coordinates ...
1
vote
0answers
149 views

Geodesic question

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the $n$-dimensional manifold of all (strictly positive) probability distributions on $\mathcal{X}=\{x_0,\dots,x_n\}$. Each $p=(p(x_0),\dots,p(x_n))\in ...
0
votes
0answers
25 views

Connection on $\operatorname{Spin}^\mathbb{C}$ spinor bundle

Let $W$ be the canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on a symplectic manifold $(M, \omega)$, with a compatible $J$ and $g$, so \begin{equation} {W_ + } = {T^{0,0}}{M^*} \oplus ...
0
votes
0answers
30 views

Question concerning $e$-geodesic

I'm learning the book on Information geometry by Amari and Nagoaka after having taken a first course on differential geometry. My question is concerning a geodesic by the $\nabla^{(e)}$-connection. ...
1
vote
0answers
53 views

Gradient Ricci soliton

I am reading Cao and Chen's paper "On Bach-flat gradient shrinking Ricci solitons". A complete Riemannian manifold $(M^n,g_{ij})$ is called a gradient shrinking Ricci soliton if there exists a smooth ...
2
votes
1answer
40 views

measure on non-oriented Riemannian manifold

Let $M$ be a non-oriented Riemannian manifold of dimension $m$. Nash embedding theorem implies that there exists an isometric embedding $\phi: M\longrightarrow \mathbb{R}^n$ for $n$ sufficiently ...
1
vote
1answer
34 views

Autoparallel submanifolds and geodesics

I have the following question in differential geometry. Any help is greatly appreciated. Let $M$ be an autoparallel submanifold of a manifold $S$ with respect to a connection $\nabla$. Let $\gamma$ be ...
1
vote
2answers
122 views

Why do people stick with Riemann-Integration when dealing with differential geometry?

I asked a question yesterday that is, "Is there an introductory differential geometry text using Lebesgue integration?" Then, i got an answer that "since we are dealing with differential geometry we ...
1
vote
1answer
65 views

Metric and Curvature on a Riemann Surface

We are given a smooth conformal metric $\rho=\rho(z)\left|dz\right|$ on a Riemann surface $X$. I have a few questions relating to this: (a) The local formula $R(\rho)=\Delta \mathrm{log}\rho dx\,dy$ ...
2
votes
0answers
70 views

Prove the Curvature Tensor is a Tensor

For an affine connection $\nabla$, prove the curvature R $R(X,Y,Z,\alpha)=\alpha(\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z -\nabla_{[X,Y]}Z)$ with $X,Y,Z$ vector fields and $\alpha$ a co-vector, is ...
10
votes
1answer
102 views

How can I understand the three-dimensional space forms?

Here is what I know: A space form is defined as a manifold admitting a Riemannian manifold of constant sectional curvature A classical result of Cartan states that a manifold is a space form if and ...
8
votes
1answer
81 views

Using index notation to write $d^2=0$ in terms of a torsion free connection.

Let $(M,g)$ be a Riemannian manifold and let $\omega$ be a $1$-form on $M$. I want to rewrite $d^2\omega=0$ in terms of the Levi-Civita connection. I can show the following: $$d\omega(X,Y) = ...
3
votes
0answers
88 views

$\operatorname{div}$ and $\operatorname{grad}$ in spherical coordinates. Formula from general relativity goes crazy

I try to calculate the gradient of a function and the divergence of a vector field in spherical coordinates. Nothing special so far, but a formula that I learned in a general relativity lecture ...
3
votes
1answer
58 views

Flatness of a manifold (or a connection)

Suppose we have an $n$-dimensional manifold $S$ (with a global coordinate system) with a metric $g$ and a connection $\nabla$ with connection coefficients (Christoffel symbols) $\Gamma_{i,j}^k$ given. ...
2
votes
1answer
58 views

Notation and hierarchy of cartesian spaces, euclidean spaces, riemannian spaces and manifolds

I am confused by some definitions. Forgive the looseness of my language. A Cartesian space is basically a space of points that can be represented by n-tuples ( and other things, but I won't go into ...
1
vote
0answers
35 views

Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic? Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...
1
vote
1answer
52 views

Parallel translation via $e$-connection

This question is concerned with Section 2.5. of Amari and Nagaoka's Information geometry book. Let me give some background first. Let $\mathcal{P}$ be the $n$-dimensional manifold of all (strictly ...
2
votes
1answer
97 views

Show isometry of flow on a compact Riemannian manifold where the vector field is Killing

Let $(M,g)$ be a Riemannian manifold, $\nabla$ the Levi-Civita connection of $g$. A vector filed $V$ on $M$ is called a Killing field if for every $p\in M$ and every $X,Y\in T_p M$, $$ g(\nabla_X V, ...
3
votes
1answer
53 views

Is this a geodesic?

Let $(M,g)$ be a riemannian manifold. Let $p$ in $M$ and $v,v_{0}$ two vectors in $\mathrm{T}_{p}M$. I am looking at the curve $$ \gamma \, : \, t \, \longmapsto \, \mathrm{Exp}_{p}(tv+v_{0}) $$ ...
2
votes
1answer
24 views

Isometric map of geodesic

Assume a Riemann manifold $(M,g)$ and a smooth map $\sigma:M\times M\rightarrow M$, $(m_{1},m_{2})\rightarrow \sigma_{m_{1}}(m_{2})$, such that: $\forall m\in M$ $\sigma_{m}:M\rightarrow M$ is an ...
1
vote
0answers
92 views

Chain rule with covariant derivative

Let $\mathcal{M}$ be a $n$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$. Consider the following function: $$\tilde{F}(v) = \operatorname{d exp}^{-1}_{p} ...
0
votes
1answer
36 views

precise meaning of connected manifold

what does it mean for a manifold to be "connected" precisely? what is the difference between a connected riemannian manifold and a nonconnected one. (i know what a riemannian manifold is a manifold ...
0
votes
1answer
50 views

connected complete totally geodesic sub manifold of $S^n$

Let $M$ and $N$ be manifolds with Riemannian metrics $g$ and $h$ respectively. A diffeomorphism $F: M\to N$ is an isometry if \begin{equation*} h_{F(x)}(T_x F(u), T_x F(v))=g_x(u,v) ...
0
votes
1answer
82 views

Covariant derivative along curve

Let $M={\mathbb R}^{3} $ with the usual metric $g=ds^{2} =dx^{2} +dy^{2} +dz^{2} $. Let $\gamma :I\to M$ be a unit speed curve. How can I prove that $\nabla _{\gamma '} \gamma '=\gamma ''$ , where ...
1
vote
1answer
35 views

Local coordinates for two riemannian metrics

Let $(M,g)$ be a Riemannian manifold, $g' = g + f$ be another metric. Is it possible to get local coordinates such that at a point $P \in M$, $g_{ij} = \delta_{ij}$ and $f_{ij} = 0$ for all $i \not = ...
1
vote
0answers
149 views

Taylor expansion of a vector field on manifold

In my work I have a need for some kind of analogue of Taylor expansion of a vector field on Riemannian manifold $\mathcal{M}$. I came to such an expression: $$ F(\operatorname{exp}_p(v)) = ...
2
votes
1answer
52 views

Gradient of a function restricted to a submanifold

Let $f$ be a sufficiently smooth function on a manifold $S$. Let $M$ be a sub manifold of $S$. Can someone show how is it then true that $(\text{grad}f|_M)_p$ at a point $p$ (gradient of the mapping ...
4
votes
1answer
70 views
+200

Topology on the space of compatible almost complex structures in symplectic geometry

I have a few fairly generic questions, with a specific application to symplectic geometry in mind. Let me pose the specific problem first: Let a symplectic manifold $(M,\omega)$ be given. One is ...
0
votes
1answer
64 views

Bishop - Gromov Comparison Theorem proof and references.

I'm having trouble understanding a proof of the Bishop's volume comparison theorem and any help would be really appreciated. It's a simple part of the proof but I'm not quite getting what they want to ...
3
votes
1answer
45 views

Hilbert-Schmidt norm/smooth manifolds

Given two riemannian manifolds $M$ and $N$ and a smooth map $f$ : $M$ $\rightarrow$ $N$, we define the energy density of $f$ as the smooth function $e(f)$ : $M$ $\rightarrow$ $\mathbb{R}$ given by ...
0
votes
1answer
69 views

How does curvature do that?

In his book "Riemannian geometry" Do Carmo said The curvature measures the amount that a riemannian manifold deviates from being euclidean My question is How does the curvature measure this ...
2
votes
2answers
45 views

On the definition/notation for pseudoholomorphic curves

A pseudoholomorphic curve is a map $u:(\Sigma,j) \to (M,J)$ from a Riemann surface $\Sigma$ with an almost complex structure $j$ to a manifold $M$ with an almost complex structure $J.$ We require ...
1
vote
0answers
46 views

Curvature tensor on the sphere

While reading R. Hamilton paper "Three-manifolds with positive Ricci curvature" I came across the sentence: For the sphere we have $$R(u,v,u,v)=R_{ijkl}u^{i}v^{j}u^{k}v^{l}>0,$$ which is the ...
1
vote
1answer
34 views

A connection is the limit of the newton quotient of the parallel transport

Let $E\rightarrow M$ be a vector bundle with connection $\nabla$. Denote by $\Pi_{\gamma(t_{0})}^{\gamma(t_{1})}:E_{\gamma(t_{0})}\rightarrow E_{\gamma(t_{1})}$ the parallel transport map along the ...
1
vote
2answers
68 views

Definition of the Energy of a curve

The energy of a curve $c: I \to S$ assuming S is a regular surface with a Riemannian metric $g$ is defined as : $$ E[c] = \frac{1}{2} \int_I g_{c(t)}(\dot c(t),\dot c(t))\mathsf{dt} $$ This is quite ...
1
vote
1answer
81 views

Lie derivative on a riemannian manifold

Suppose we have a Riemannian manifold $(M,g,\nabla)$ with Levi-civita connection $\nabla$. We define a new symmetric non-metric connection $\bar\nabla$ on $M$. Then the Lie derivative of functions and ...
1
vote
1answer
28 views

Any books on isospectral manifolds?

I was searching stuff related to M.Kac's famous question "Can one hear the shape of the drum ?" I further found results due to Gordon, Webb and Wolpert in the 2D case using Sunada method. Are there ...
4
votes
1answer
117 views

What does it mean “being geodesic” is not invariant?

We know that "being geodesic" is not invariant under re-parametrization. Only affine re-parametrization preserves the property of being a geodesic. Also, a geodesic is locally distance minimizer. ...
1
vote
1answer
105 views

A new symmetric non-metric connection that generalizes the geodesic equation(Version 2)

A curve $\alpha$ on a riemannian manifold $(M,g,\nabla)$ is a geodesic if $\nabla_TT=0$, where $T$ is the tangent vector field. A generalization of this geodesic equation suggests that $\nabla_TT=\rho ...